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A rigorous way to define the sine function is to consider it as the solution to the IVP: $$ \begin{cases} y^{\prime \prime} + y = 0\\ y(0) = 0 \\ y^{\prime}(0) = 1 \end{cases} $$


Some other important formulas regarding $\sin(x)$ that you didn't mention are the infinite product $$\sin(x) = x \prod_{k=1}^{\infty}\Big( 1 - \frac{x^2}{\pi^2 k^2} \Big)$$ and the partial fractions decomposition $$\frac{1}{\sin(x)^2} = \sum_{k=-\infty}^{\infty} \frac{1}{(x-\pi k)^2}, \; \; x \notin \pi \cdot \mathbb{Z},$$ although I guess the latter only ...


Expanding my comments into an answer: by distributing the divisions by $a_0$, $a_1$, $\ldots$ successively you can rewrite such an upward continued fraction in the equivalent form $\frac1{a_0}+\frac1{a_0a_1}+\frac1{a_0a_1a_2}+\cdots$. This is known as the Engel expansion of the number, and their coefficients have some interesting limiting properties (in ...

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